You are thinking far too much about biological neurons as CPUs or some similar electronic equivalent; they are not, they are biological machines and much of the fidelity of biological signal processing comes from massively parallel probabilistic networks with a lot of simultaneous redundancy, rather than serial processing of a stack. Neurons encode information in space (which neurons are firing) and time (when do those neurons fire). There is no biological mechanism for encoding the strength of a signal beyond the maximum firing rate of an individual neuron without considering the ability of a group of neurons to fire together.
If you are interested in biological neurons, I would strongly suggest starting from a basic neuroscience textbook. You will get a lot further learning bottom-up rather than trying to reverse engineer what you know about artificial neural networks and network models in thinking about biology. Two books I typically suggest (any edition of either is probably fine) are Neuroscience by Dale Purves, and Principles of Neural Science by Eric Kandel. You can consider either of those books as references for the rest of this answer, which is all very basic in the context of a first-semester neuroscience course.
Neurons in space
Neurons are not single-compartments (whereas most artificial neural network units are); they extend across non-isoelectric space, so events occurring at the soma/axon hillock are filtered before affecting the dendrites (where inputs tend to arrive). Therefore, "what happens during the refractory period" depends very much on where you are in the cell. This is addressed a bit in this Q&A: Does an action potential abolish an excitatory postsynaptic potential?
When neurotransmitters bind ligand-gated ion channels, a conductance is opened with some reversal potential. What happens to the membrane next depends on other parameters, in particular the magnitude of other open conductances (or the inverse, the input resistance) and the membrane voltage. Excitatory conductances tend to have reversal potentials near 0 mV, so the closer the cell is to 0 mV the smaller the evoked current will be. During action potentials (at least near the soma, and possibly propagated into the dendrites), the cell is depolarized and has low input resistance due to open voltage-gated channels. Inputs that arrive in that context are indeed going to be blunted if not entirely "wiped out".
Absolute refractory period
There are two relevant refractory periods to consider. One called the "absolute" refractory period refers to the period of time during and immediately after an action potential. Action potentials occur due to a positive feedback loop of opening voltage-gated (typically sodium) channels. After opening, these channels inactivate and cannot be reopened for some time. This inactivation is necessary for an efficient repolarization of the cell back near rest after an AP is fired. These channels need to recover sufficiently from inactivation in order to fire again.
Relative refractory period
After the absolute refractory period, there is an additional interval in which a cell is more hyperpolarized than normal. This state is caused by the voltage-gated potassium channels that are crucial for repolarizing the cell quickly, with the trade-off that the cell is further from threshold for a brief time. The relative refractory period can be overcome with a sufficiently large stimulus, whereas the absolute refractory period cannot.